If-f-x-2-x-86-and-g-x-3x-2-x-4-Then-find-g-f-1-g-14-

Question Number 200168 by hardmath last updated on 15/Nov/23

If f (x) = 2^{x} + 86 and g (x) = {3 x}^{2} + x - 4

Then find : g [f^{- 1} (g (14))] = ?

Commented by jazeee last updated on 15/Nov/23

S o l u t i o n :

W e n e e d t o f i n d f^{- 1} (x) f i r s t .

y = 2^{x} + 86

x = 2^{x} + 86

x - 86 = 2^{y}

l n (x - 86) = y * l n (2)

\frac{l n (x - 86)}{l n (2)} = y

f^{- 1} (x) = \frac{l n (x - 86)}{l n (2)}

N o w, s u b s t i t u t e 14 i n g (x) .

g (14) = 3 {(14)}^{2} + (14) - 4

g (14) = 3 (196) + 10

g (14) = 588 + 10

g (14) = 598

N e x t i s t o s u b s t i t u t e 598 i n f^{- 1} (x) .

f^{- 1} (598) = \frac{l n (598 - 86)}{l n (2)}

f^{- 1} (598) = \frac{l n (512)}{l n (2)}

n o t e : 512 = 2^{9}

f^{- 1} (598) = \frac{9 l n (2)}{l n (2)}

f^{- 1} (598) = 9

L a s t l y, s u b s t i t u t e 9 i n g (x) .

g (9) = 3 {(9)}^{2} + (9) - 4

g (9) = 3 (81) + 5

g (9) = 243 + 5

g (9) = 248

A n s w e r : 248 .

Commented by jazeee last updated on 15/Nov/23

* x = 2^{y} + 86 * (s e c o n d s t e p)

Answered by Sutrisno last updated on 15/Nov/23

g (14) = 3 {(14)}^{2} + 14 - 4 = 598

f (x) = 2^{x} + 86 \to x = f^{- 1} (2^{x} + 86)

2^{x} + 86 = 598

2^{x} = 512 \to x = 9

g (9) = 3 {(9)}^{2} + 9 - 4 = 248

∴ g (f^{- 1} (g (14))) = 248

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